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Least integer of each prime signature, in graded (reflected or not) colexicographic order of exponents.
37

%I #51 Oct 19 2020 23:13:41

%S 1,2,4,6,8,12,30,16,24,36,60,210,32,48,72,120,180,420,2310,64,96,144,

%T 216,240,360,900,840,1260,4620,30030,128,192,288,432,480,720,1080,

%U 1800,1680,2520,6300,9240,13860,60060,510510,256,384,576,864,1296,960,1440,2160

%N Least integer of each prime signature, in graded (reflected or not) colexicographic order of exponents.

%C The exponents can be read off Abramowitz and Stegun, p. 831, column labeled "pi".

%C Here are the partitions in the order used by Abramowitz and Stegun (graded reflected colexicographic order): 0; 1; 2, 1+1; 3, 1+2, 1+1+1; 4, 1+3, 2+2, 1+1+2, 1+1+1+1; 5, 1+4, 2+3, 1+1+3, 1+2+2, 1+1+1+2, 1+1+1+1+1; ... (Cf. A036036)

%C Here are the partitions in graded colexicographic order: 0; 1; 2, 1+1; 3, 2+1, 1+1+1; 4, 3+1, 2+2, 2+1+1, 1+1+1+1; 5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1; ... (Cf. A036037)

%C Since the prime signature is a partition of Omega(n), so to speak, the internal order is only a matter of convention and has no effect on the least integer with a given prime signature.

%C The graded colexicographic order has the advantage that the exponents are in the same order as the least integer with a given prime signature (also used on the wiki page, see links).

%C Embedded values include the primorial numbers 1, 2, 6, 30, 210, 2310, 30030 ... (A002110) with unordered factorizations counted by A000110 (Bell numbers) and ordered factorizations by A000670 (ordered Bell numbers).

%C When viewed as a table the n-th row has partition(n) (A000041(n)) terms. - _Alford Arnold_, Jul 31 2003

%C A closely related sequence, A096443(n), gives the number of partitions of the n-th multiset. - _Alford Arnold_, Sep 29 2005

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings).

%H Peter Luschny, <a href="/A036035/b036035.txt">Rows n = 0..25, flattened</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://people.math.sfu.ca/~cbm/aands/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.

%H John Baez, <a href="https://plus.google.com/u/0/117663015413546257905/posts/XAokddKJEBt">What happens when a particle gets created?</a>

%H OEIS Wiki, <a href="http://oeis.org/wiki/Prime_signature">Prime signature</a>.

%e 1;

%e 2;

%e 4, 6;

%e 8, 12, 30;

%e 16, 24, 36, 60, 210;

%e 32, 48, 72, 120, 180, 420, 2310;

%e 64, 96, 144, 216, 240, 360, 900, 840, 1260, 4620, 30030;

%e 128, 192, 288, 432, 480, 720, 1080, 1800, 1680, 2520, 6300, 9240, 13860, 60060, 510510;

%p with(combinat):

%p A036035_row := proc(n) local e, w; w := proc(e) local i, p;

%p p := [seq(ithprime(nops(e)-i+1), i=1..nops(e))];

%p mul(p[i]^e[i], i=1..nops(e)) end:

%p seq(w(conjpart(e)), e = partition(n)) end:

%p seq(A036035_row(i), i=0..10); # _Peter Luschny_, Aug 01 2013

%t nmax = 52; primeSignature[n_] := Sort[ FactorInteger[n], #1[[2]] > #2[[2]] & ][[All, 2]]; ip[n_] := Reverse[ Sort[#]] & /@ Split[ Sort[ IntegerPartitions[n], Length[#1] < Length[#2] & ], Length[#1] == Length[#2] & ]; tip = Flatten[ Table[ip[n], {n, 0, 8}], 2]; a[n_] := (sig = tip[[n+1]]; k = 1; While[sig =!= primeSignature[k++]]; k-1); a[0] = 1; a[1] = 2; Table[an = a[n]; Print[an]; an, {n, 0, nmax}](* _Jean-François Alcover_, Nov 16 2011 *)

%o (PARI) Row(n)={[prod(i=1, #p, prime(i)^p[#p+1-i]) | p<-partitions(n)]} \\ _Andrew Howroyd_, Oct 19 2020

%Y A025487 in a different order. Cf. A035098, A002110, A000110 and A000670.

%Y Cf. A025487, A059901, A096443.

%K nonn,easy,nice,tabf,look

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Alford Arnold_; corrected Sep 10 2002

%E More terms from _Ray Chandler_, Jul 13 2003

%E Definition corrected by _Daniel Forgues_, Jan 16 2011